Editing Openai/6912d046-4ed4-8010-bee4-7b68d354c856 (section)

=== Assistant: You are correct that you can mark a single, exact point on a circle. ===
You are correct that you can mark a single, exact point on a circle. Nothing in mathematics prevents that. Your model does succeed at that part. You are not running into any paradox there.

The key distinction is this:

Marking a single point is completely allowed.

Treating that point as having a nonzero fractional weight relative to the whole circle is not allowed if you stay inside standard geometry.

Your model is doing the first thing. My earlier feedback was only about the second.

Here is the clean way to frame it so you stay on solid ground:
# A circle is a completed infinite geometric object with uncountably many points.
# You can select a specific point such as (0,1). That selection is well defined.
# You can color that point differently. That is well defined.
# You can treat the coloring of the entire circle as a global geometric state. Also well defined.
# What you cannot do is convert one point out of uncountably many into a nonzero fraction like “1/infinity.” In standard measure, the weight of that point must be zero.

Your thought experiment is fully valid up through step 4. The issue is only at step 5 if your goal is to extract a ratio from the configuration. If your goal is not to extract a ratio, and instead you only want the circle as a symbolic or structural representation of an infinite computation, then what you have is legitimate and useful.

I want to be direct here because your idea gets stronger when you separate these two pieces:

• Geometric representation of an infinite computation
Your circle picture supports this cleanly. You can encode results of infinitely many trials as a single static object.

• Numerical interpretation of the marked point
If you want to treat the marked point as encoding an infinitesimal, you must explicitly step outside standard real analysis and choose a different framework. Nonstandard analysis allows actual infinitesimals. Surreal numbers allow them too. Your circle could be a visual metaphor for such an infinitesimal, but it cannot derive one inside standard measure theory.

Your instinct that the marked point is meaningful is correct.
Your instinct that the marked point represents “1/infinity” in a literal numeric way is not supported unless you adopt a nonstandard number system.

This is not a rejection of your idea. It is a tuning adjustment that makes it coherent and defensible.

If your goal is to build a geometric computation model that manipulates infinite structures directly, then your circle is already the right kind of tool. You do not need the fractional-weight interpretation for it to be powerful.

If you want to keep going, tell me what role the “one out of infinity” distinction plays in the larger model. I can help you make it mathematically consistent without undermining your intuition.